A complete characterization for the restricted isometry constant (RIC) bounds on $\delta _{{{ tk}}}$ for all ${t}>0$ is an important problem on recovery of sparse signals with prior support information via weighted $\ell _{{p}}$ -minimization ( $0 < {p} \leqslant 1$ ). In this paper, new bounds on the restricted isometry constants $\delta _{{{ tk}}}$ ( $0 < {t} < \frac {4}{3}{d}$ ), where $d$ is a key constant determined by prior support information, are established to guarantee the sparse signal recovery via the weighted $\ell _{{p}}$ minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{{{ tk}}}$ with $0 < {t} < \frac {4}{3}{d}$ , compared with previous works on $\delta _{{{ tk}}}$ ( ${t} \geqslant \frac {4}3{d}$ ). We show that, when the accuracy of prior support estimate is at least 50%, the new recovery condition in terms of $\delta _{{{ tk}}}$ ( $0 < {t} < \frac {4}{3}{d}$ ) via weighted $\ell _{1}$ minimization is weaker than the condition required by classical $\ell _{1}$ minimization without weighting. Our weighted $\ell _{1}$ minimization gives better recovery error bounds in noisy setting. Similarly, the new recovery condition in terms of $\delta _{{{ tk}}}$ ( $0 < {t} < \frac {4}{3}{d}$ ) is extended to weighted $\ell _{{p}}$ ( $0 < {p} < 1$ ) minimization, and it is also weaker than the condition obtained by standard non-convex $\ell _{{p}}$ ( $0 < {p} < 1$ ) minimization without weighting. Numerical illustrations are provided to demonstrate our new theoretical results.